A harmonizer is a pitch shifter, but not all pitch shifters are harmonizers. You keep saying all pitch shifters use an internal pitch detector, but that doesn't appear to be the case. Otherwise they couldn't be used for complex chords, complex sounds with no natural harmonic series, or any other fully polyphonic material, without limits. I can pitch shift an entire orchestral recording and there's no pitch to detect in that. They just chop the audio up into windows, drop data or duplicate data, stretch or compress the audio in the window, then merge the windows back together. No pitch detection needed.
I understand that even monophonic pitch shifters can sometimes handle octaves or other notes that fall onto the natural harmonic series of the fundamental just as my two monophonic pitch shifters can. But as soon as a complex chord comes in they screw up as you say. But that's not the case with the non-pitch detecting pitch shifters in my effects units by companies such as Roland, Boss, Lexicon, TC Electronics, Behringer, and Digitech. My Boss SX-700 has 5 non-pitch detecting modes that can handle any kind of input and 3 mono pitch detecting modes for monophonic input, and the user can select whether to detect the pitch of the actual audio that will be pitch shifted or the user can provide another pitch in another channel for the unit to detect. So I think I know what the differences are.
I have a feeling the reason this keeps going on and on is that you don't want to admit that the Eventide doesn't do something that most other effects units can. But it's important not to hide or confuse that information, because if I forked out several thousand dollars on an Eventide, only to find out it can't do one of the main things I bought it for, it would probably get boxed up and sent back and I would not be happy.